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Properties

Label 20.8.19751718904...5625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{14}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}$
Root discriminant $36.71$
Ramified primes $5, 61, 97, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 183, -2946, 11997, -24879, 31442, -27658, 19708, -13045, 7139, -1690, -1050, 1545, -1084, 311, 31, -71, 35, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 35*x^17 - 71*x^16 + 31*x^15 + 311*x^14 - 1084*x^13 + 1545*x^12 - 1050*x^11 - 1690*x^10 + 7139*x^9 - 13045*x^8 + 19708*x^7 - 27658*x^6 + 31442*x^5 - 24879*x^4 + 11997*x^3 - 2946*x^2 + 183*x + 41)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 + 35*x^17 - 71*x^16 + 31*x^15 + 311*x^14 - 1084*x^13 + 1545*x^12 - 1050*x^11 - 1690*x^10 + 7139*x^9 - 13045*x^8 + 19708*x^7 - 27658*x^6 + 31442*x^5 - 24879*x^4 + 11997*x^3 - 2946*x^2 + 183*x + 41, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} + 35 x^{17} - 71 x^{16} + 31 x^{15} + 311 x^{14} - 1084 x^{13} + 1545 x^{12} - 1050 x^{11} - 1690 x^{10} + 7139 x^{9} - 13045 x^{8} + 19708 x^{7} - 27658 x^{6} + 31442 x^{5} - 24879 x^{4} + 11997 x^{3} - 2946 x^{2} + 183 x + 41 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19751718904610600919732666015625=5^{14}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.71$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 61, 97, 397$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{263} a^{18} + \frac{58}{263} a^{17} + \frac{58}{263} a^{16} + \frac{41}{263} a^{15} - \frac{16}{263} a^{14} - \frac{72}{263} a^{13} + \frac{116}{263} a^{12} - \frac{66}{263} a^{11} + \frac{24}{263} a^{10} - \frac{109}{263} a^{9} - \frac{130}{263} a^{8} - \frac{124}{263} a^{7} + \frac{110}{263} a^{6} - \frac{54}{263} a^{5} + \frac{105}{263} a^{4} - \frac{2}{263} a^{3} - \frac{129}{263} a^{2} + \frac{60}{263} a + \frac{84}{263}$, $\frac{1}{2703652484272405501809707585979227} a^{19} + \frac{2124504750774730787727798818828}{2703652484272405501809707585979227} a^{18} - \frac{430849089071411369804284895517304}{2703652484272405501809707585979227} a^{17} - \frac{171795309091112657013319620323052}{2703652484272405501809707585979227} a^{16} - \frac{85667111244378551696320558461634}{2703652484272405501809707585979227} a^{15} + \frac{1218075772281430488395049075841725}{2703652484272405501809707585979227} a^{14} + \frac{964955505256079037841618789162938}{2703652484272405501809707585979227} a^{13} + \frac{257536499055766543549368019370052}{2703652484272405501809707585979227} a^{12} - \frac{75539758791271399454823268267}{10280047468716370729314477513229} a^{11} - \frac{46855461281039027785661170442185}{2703652484272405501809707585979227} a^{10} - \frac{564568101179000380052636401771581}{2703652484272405501809707585979227} a^{9} - \frac{759734925257177103468064458033880}{2703652484272405501809707585979227} a^{8} + \frac{1178926090663919476710373419063109}{2703652484272405501809707585979227} a^{7} + \frac{300089296751756977802943874230331}{2703652484272405501809707585979227} a^{6} - \frac{1139913134925163095738282581211575}{2703652484272405501809707585979227} a^{5} + \frac{329118204451721363957758855633562}{2703652484272405501809707585979227} a^{4} - \frac{130652806070453378823082673286253}{2703652484272405501809707585979227} a^{3} - \frac{526994010343919936514755837251045}{2703652484272405501809707585979227} a^{2} + \frac{267374011477490889778092417954187}{2703652484272405501809707585979227} a + \frac{572233318819771305090207665378512}{2703652484272405501809707585979227}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 147520359.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
61Data not computed
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
397Data not computed